Japt, 7 6 bytes

å+ ¬¬x

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Explanation

å+ ¬¬x   Implicit: input = string
å+       Cumulative reduce by concatenation. Gives the list of prefixes.
   ¬     Join into a single string of digits.
    ¬    Split back into digits.
     x   Sum.
         Implicit: output result of last expression

Japt, 7 6 4 bytes

å+ x

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Explanation

å+ x    Implicit: input = digit list
å+      Cumulative reduce by addition. Gives the sum of each prefix.
   x    Sum.

Old solution:

å+ ¬¬x

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Explanation

å+ ¬¬x   Implicit: input = string
å+       Cumulative reduce by concatenation. Gives the list of prefixes.
   ¬     Join into a single string of digits.
    ¬    Split back into digits.
     x   Sum.
         Implicit: output result of last expression

Japt, 7 bytes

å+ x_¬x

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Japt, 7 6 bytes

å+ ¬¬x

Try it online!

Explanation

å+ ¬¬x   Implicit: input = string
å+       Cumulative reduce by concatenation. Gives the list of prefixes.
   ¬     Join into a single string of digits.
    ¬    Split back into digits.
     x   Sum.
         Implicit: output result of last expression